Nematic liquid crystal macroscopic theory

The macroscopic theory of elasticity can explain why longitudinal ridges and troughs on the surface of a glass are conducive to the planar homogeneous alignment of nematic liquid crystals [23]. For simplicity, a sinusoidal shape is chosen for the cross-section of a surface relief with the wavevector q directed along x, see Fig. 10.21a ... 

Continuum theory generally employs a unit vector field n(x) to describe the alignment of the anisotropic axis in nematic liquid crystals, this essentially ignoring variations in degrees of alignment which appear to be unimportant in many macroscopic effects. This unit vector field is frequently referred to as a director. In addition, following Oseen [1] and Frank [4], it commonly assumes the existence of a stored energy density W such that at any point... 

An important aspect of the macroscopic structure of liquid crystals is their mechanical stability, which is described in terms of elastic properties. In the absence of flow, ordinary liquids cannot support a shear stress, while solids will support compressional, shear and torsional stresses. As might be expected the elastic properties of liquid crystals are intermediate between those of liquids and solids, and depend on the symmetry and phase type. Thus smectic phases with translational order in one direction will have elastic properties similar to those of a solid along that direction, and as the translational order of mesophases increases, so their mechanical properties become more solid-like. The development of the so-called continuum theory for nematic liquid crystals is recorded in a number of publications by Oseen [ 1 ], Frank [2], de Gennes and Frost [3] and Vertogen and de Jeu [4] extensions of the theory to smectic [5] and columnar phases [6] have also been developed. In this section it is intended to give an introduction to elasticity that we hope will make more detailed accounts accessible the importance of elastic properties in determining the... 

Among the characteristic physical properties of liquid crystals, what are of critical importance to display devices (LCDs) are those of macroscopic spatiotemporal scale there, the theories of liquid crystals as continuous media play essential roles. The basis of static continuum mechanics of nematic liquid crystals was established by... 

The theoretical description of the nematic liquid crystals is based on the macroscopic theory of elasticity. It introduces an order parameter that is zero in the isotropic phase and nonzero in the nematic, and can be directly related to macroscopic quantities, such as the anisotropy of the magnetic susceptibility. Because of the symmetry properties of the uniaxial nematic phase, the order parameter Q is a traceless tensor. Defining the director n as a unit vector parallel to the local average orientation of the elongated molecules, the tensor order parameter Q can be written as... 

Within the mean-field theory there are no spontaneous elastic deformations since any deformation increases the free energy. However, when a nematic LC is subject to interactions with the confining walls the homogeneous order can be perturbed. On the microscopic level, the molecules of the walls and of the liquid crystal attract each other via a short-range van der Waals interaction. In the macroscopic description this is modeled with a contact quadruple-quadruple interaction, known as the Rapini—Papoular model [32,33], which to the lowest order reads... 

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